Transport Window for Bunching Beams in a Final Stage of Heavy Ion Inertial Fusion

Tokyo Institute of Technology, Department of Energy Sciences

Takashi KIKUCHI, Mitsuo NAKAJIMA and Kazuhiko HORIOKA

International Workshop on Recent Progress of Induction Accelerators (RPIA2002), October 29-31, 2002, Tsukuba, Japan

Background & Purpose

Intense Heavy Ion Beam (~10GeV ~10ns ~100kA) Generation & Transport are required for effective implosion.

Heavy Ion Inertial Fusion

BeamBunching

Induction Linear Accelerator

Final Beam Bunching

~100ns ~10ns

Accelerated Heavy Ion Beam

Focusing

Beam

Beam Irradiation for Implosion

Dynamics of Bunching BeamBunching Beam Transportability in Phase Advancing Condition

Researches:

Heavy Ion Beam Bunching by Induction LINAC

Bipolar Voltage Waveform

Apply Bunching Voltage at each Gapfor beam bunching

∆ββ

head

tail

Beam

t

V

Induction buncher consists of periodic lattice, acceleration gaps & FODO quadrupole.

apply to beam

Head & tail velocities are modulated. ∆β/β indicates Velocity Tilt.

Beam head is deceleratedBeam tail is accelerated

by

Beam bunch becomes short during transport.

Velocity Tilt Induced Instability

Head-to-Tail Velocity Tilt for Bunch Compression

Transverse Mismatch in terms of Quadrupole Lens

Beam Transport Instability

Prediction Resonance Effect by Phase Advance Modulation

Verify Simple Estimation of Instability using Particle-Core Model

Envelope OscillationResonance Instability

Beam Parameters

Ion SpeciesIon Number

Total ChargePulse Duration

Total Beam CurrentBeam Number

Current per BeamParticle Energy

Longitudinal Beam Length

Pb1+ (207.2 amu)2.5x1015

0.4mC250ns ⇒ 10ns1.6kA ⇒ 40kA

4400A ⇒ 10kA

10GeV ( β ~ 0.31 )

23m ⇒ 0.9m

by J.J. Barnard, et al., Physics of Fluid B 5, 2698 (1993).

Parameters in final stage of HIF driver system

Bunch Compression Ratio = 25

Envelope Calculation Result with Bunching

Transverse Behavior

Beam bunching causes current (perveance) increase.Radius Expansion

Longitudinal BehaviorBunching voltages apply velocity tilt to beam,bunch compression is caused.

Core/Particle Resonance

YX

Beam Core Oscillation

Particle Oscillations around CoreResonance

x

y

ωcore / ωparticle=integer

by J.-M. Lagniel, Nucl. Instrum. Methods Phys. Res. A345 (1994) 46.

Halo formation, larger orbit of particle, chaotic behavior, etc. are induced by resonance.

Beam Core

Halo Particles

Poincaré Maps

σ0=60˚ ∆β/β=0 σ0=60˚ ∆β/β=5%

Lager Orbitby Resonance

Particle Loss & Halo Formation

x [cm]

dx/d

s [m

rad]

x [cm]

Resonance reentrancedue to ∆β/β

Expression of Phase Advance in FODO Lattice

( ) θθθθθθθθθθσ sinhsin21coshsinsinhcoscoshcoscos 2

2

0

−−+=ll

LL

σo : Phase Advance without Space Charge [deg]θ : Focusing Strength of Lens [rad]L : Length of Drift Space (O in FODO) [m]ℓ : Length of Focusing Space (F or D in FODO)

from Transfer Matrix

F

ℓ

s

L

O ODkt

2

0 2coscos

++=

aLK lσσ

Calculate σ0

σ : Phase Advance Depressed by Space Charge

Estimate σ

( )σ

εsin

22 La t += l

Average Beam Radius

2

=l

θkℓ

ℓ ℓ

ℓ ℓ

Beam Envelope Calculation

Transverse Envelope Equation (K-V Distribution)

modulated by ∆β/β

3

2

2

2 ~2~XYX

KXkds

Xd tt

ε++

+−=

3

2

2

2 ~2~YYX

KYkds

Yd tt

ε++

+=

)(sXX ≡

γβ ~~1~ ∝tk

s : Transport Distance

32 ~~1~γβ

∝K

Velocity Tilt induces Beam Envelope Mismatch

3

2'

22

2

ZZKk

dsZd zzL

zε++−= = 0

Longitudinal Envelope Equationtk : Transverse Focusing Force: Transverse Perveance

zk : Longitudinal Focusing ForceLK : Longitudinal Perveance

K

)(sYY ≡ )(sZZ ≡

Transport Window Shrinking with Compression

Transport Window becomes small with longitudinal compression (K increase).

For Avoiding the Resonance Conditions ...

It is difficult that entire beam stability is maintained during compression.

Velocity Tilt ∆β/β

Phas

e A

dvan

ce σ

Phase Advance Dynamics with Compression

K/K0=1

K/K0=2

K/K0=3

from Envelope Calculation

at Z/3

The phase advance goes off Transport Window during compression.

Particle-Core Model (PCM) Calculation

Test Particle Equation in PCM

Beam Core

x

y

Test Particle

Beam cross section

calculated by Env. Eqs.

( ) ( )

( )

>−++

⋅

≤+

+−=Xx

XYxxKx

XxxYXX

K

xkds

xdx

222

2

2

~2)sgn(

~2

~

Poincaré plot at each slice with compression

–1 –0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

z/Z

λ/λ

0

(∆β

/β) / (∆

β/β

)max

123456

1 2 3

4 5 6From relation between velocity tilt & perveance, resonance effect is appeared at near center, rather than beam head.

PCM calculations are carried out at each sliced position.Each Poincaré plot is shown in

Summary

Resonance reentrance due to ∆β/β

Instability Growth

Simple Prediction of Phase Advance Modulation

Resonance Reentrance & Envelope Mismatch

decide Transport Window for Bunching Beam

Transport Window changes with Longitudinal Compression